It is not permitted at all for a man to shake hands with a non-Mahram woman.

It is not permissible for a man to touch the hands of a woman who is not one of his Mahrams. It is not permissible to shake hands even with a barrier in between such as shaking hands from beneath a garment and the like. The same ruling applies to shaking hands with old women.

How to consummate marriage: 1: you should treat your bride kindly and gently; 2- you should place your hand on your bride’s head and pray for her; 3- you should pray two rak’ahs with her; and 4- you should say the du’a reported by Ibn 'Abbas.

There is nothing wrong with responding when a non-Muslim greets a Muslim at Eid or on any other occasion, by wishing him well and praying for him in an appropriate manner that is equivalent to his greeting. So one may say to him, “May Allah help you” or “May Allah honour you” and the like, and if the intention is to pray that Allah bless him and guide him to Islam, then that is more appropriate and better. And Allah knows best.